{"paper":{"title":"Complexity of short rectangles and periodicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bryna Kra, Van Cyr","submitted_at":"2013-06-29T12:52:35Z","abstract_excerpt":"The Morse-Hedlund Theorem states that a bi-infinite sequence $\\eta$ in a finite alphabet is periodic if and only if there exists $n\\in\\N$ such that the block complexity function $P_\\eta(n)$ satisfies $P_\\eta(n)\\leq n$. In dimension two, Nivat conjectured that if there exist $n,k\\in\\N$ such that the $n\\times k$ rectangular complexity $P_{\\eta}(n,k)$ satisfies $P_{\\eta}(n,k)\\leq nk$, then $\\eta$ is periodic. Sander and Tijdeman showed that this holds for $k\\leq2$. We generalize their result, showing that Nivat's Conjecture holds for $k\\leq3$. The method involves translating the combinatorial pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0098","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}